For a dairy free alternative, try using soy milk, almond milk, or
coconut milk (rice milk will make your baked goods too dry, so try to
avoid it in baking). When a recipe calls for milk, I always like to use
whole milk or buttermilk, for a richer, more moist and tender baked good
but 2% works too.

There are many egg substitutes in baking, eggs perform a different function, depending on what you're baking. The main functions of eggs in baking are thickening, binding to hold the ingredients together and leavening to make baked goods rise light and fluffy. Identifying their function in a particular recipe will help you decide with how to replace them. Different egg replacers will work best in different recipes. For example, in custard pies, like pumpkin pie, eggs are mainly for thickening. In quick breads, cakes and cupcakes, both leavening and binding is needed, while extra yolks add richness and extra whites help to make the cake light and fluffy. Yet, in cookies, the egg acts as a binder, holding the ingredients together and adding moisture. Note, I don't recommend using egg substitutes with recipes that use a lot of eggs, egg yolks, or recipes that require a lot of egg white like angel food cake. The less egg in any recipe, the easier to substitute and less noticeable it will be. Egg substitutes like Ener-G Egg Replacer are probably the most simple and effective, but here are some simple alternatives from ingredients you might already have in your kitchen.

Substitute 1 large egg with:

1 tablespoon vinegar, white or apple cider in cakes & cupcakes

1 tablespoon ground flax seed meal, mixed with 2 tablespoons water and let sit for a few minutes to thicken, in cookiesor brownies

It is always best to substitute a solid fat for a solid fat (butter, margarine, shortening, lard) and a liquid fat for a liquid fat (canola oil, vegetable oil, etc.). Solid fats that have been melted do not become a "liquid fat". Butter for instance, contains milk solids (oil does not) that will firm and set after your baked good has cooled.

Butter: Unsalted butter is usually called for when not specifically stated but salted butter can be used in its place by reducing the amount of salt in the recipe by 1 teaspoon per pound of butter. I do not recommend substituting oil for butter or using low-fat spreads, margarine or light butter for baking.

Oil: Substitute half of the oil in the recipe for unsweetened applesauce to cut the fat for a healthier alternative. For example, if the recipe calls for 1/4 cup oil, use 2 tablespoons oil and 2 tablespoons applesauce. Note this may change the texture of your baked goods, and make them more dense and chewy.

Chocolate: Do not substitute chocolate syrup for melted chocolate OR cocoa/hot chocolate mix for cocoa powder in any recipe. These substitution will not work as well as the original chocoalte needed in all recipes.

1 ounce bittersweet chocolate = 1 ounce semisweet chocolate; Bittersweet and semisweet chocolate may be used interchangeably, but note that it will change the flavor and texture slightly, as semisweet chocolate is more sweet than bittersweet chocolate.

1 ounce milk chocolate = 1 ounce semisweet chocolate; milk chocolate and semisweet chocolate may be used interchangeably, but note that it will change the flavor and texture slightly, as milk chocolate is more sweet than semisweet chocolate.

Dutch-Process Cocoa Powder: May substitute equal amounts of natural unsweetened cocoa powder plus 1/8 teaspoon baking soda per every 1/4 cup. This
will neutralize the acids in the natural cocoa, so you are able to use
it in a recipe that relies on baking powder for leavening. (See ingredients 101 for more info on types of cocoa powder)

Natural Unsweetened Cocoa Powder: May substitute equal amounts of Dutch-processed cocoa powder, omitting any baking soda called for in the recipe.

Leavening

Baking Soda: There is no recommended homemade substitute for baking soda.

Baking Powder: To substitute 1 teaspoon of single acting baking powder, combine 1/4 teaspoon baking soda mixed
with 1/2 teaspoon cream of tartar and 1/4 teaspoon cornstarch. There is
no recommended substitute for double-acting baking powder, most
commonly sold in grocery stores. (See ingredients 101 for more info on the difference between the two)